Summary

Morphogenesis in living organisms relies on the integration of both biochemical and mechanical signals. During the last decade, attention has been mainly focused on the role of biochemical signals in patterning and morphogenesis, leaving the contribution of mechanics largely unexplored. Fortunately, the development of new tools and approaches has made it possible to re-examine these processes. In plants, shape is defined by two local variables: growth rate and growth direction. At the level of the cell, these variables depend on both the cell wall and turgor pressure. Multidisciplinary approaches have been used to understand how these cellular processes are integrated in the growing tissues. These include quantitative live imaging to measure growth rate and direction in tissues with cellular resolution. In parallel, stress patterns have been artificially modified and their impact on strain and cell behavior been analysed. Importantly, computational models based on analogies with continuum mechanics systems have been useful in interpreting the results. In this review, we will discuss these issues focusing on the shoot apical meristem, a population of stem cells that is responsible for the initiation of the aerial organs of the plant.

I. Introduction

Since Aristotle, who published one of the first known classifications of living beings, the natural sciences have tried to sort the diversity of shapes found among living organisms. Building on these descriptive studies, developmental biologists have tried to understand how the organisms acquire their mature shape. This question can be comprehended by considering that morphogenesis, the generation of shape, can be described using simple geometry. In this framework, the focus switches to the analysis of how a system generates patterns of growth following basic geometrical rules.

A precise geometrical description is, however, not sufficient to understand how shapes arise. In an attempt to unravel the mechanisms behind this process, developmental biologists have turned to molecular genetics and the past decades have seen an impressive accumulation of genetic data identifying the molecules that are at the basis of developmental processes. The wealth of information has led to a concept where most if not all of the patterning processes are largely controlled by chemical regulation. Shape, however, cannot be described in terms of chemistry alone. Indeed, the link between geometry and morphogenesis necessarily involves physical processes and the development of all organisms is limited by mechanical constraints. For this reason attention has also been focused on the link between biological and mechanical processes providing strong correlation-based evidence for the control of growth and development by physical constraints. This biomechanical approach of development has been established in plants by Simon Schwendener (Schwendener, 1874, 1878) and later developed by Thomas D’Arcy Wentworth Thompson (D’Arcy Thompson, 1917) in his book On Growth and Forms and by Paul Green during his entire career (Green, 1999). However, because this work has been technically very challenging, it has been difficult to go beyond correlations and only recently has the role of mechanical forces during development been re-examined, notably by using modern live imaging techniques coupled with systems biology approaches.

In this review we will present, from the biologists’ perspective, our current understanding of how shape is determined during plant development. We will focus our discussion on meristematic tissues, through which most of plant architecture is determined.

II. The cellular basis of plant morphogenesis and plant mechanics

In plants (as in bacteria and fungi), osmotic water uptake causes the cells to become turgid. The internal pressure, or turgor pressure, makes the cells swell and it is only the presence of a stiff exoskeleton, the cell wall, that prevents them from bursting. In growing plant cells, cell wall synthesis and remodeling causes the external matrix to yield to the turgor pressure, thus allowing the cell to grow. Turgor pressure is thus often considered as the motor of growth.

Pfeffer was the first to measure osmotic pressure indirectly (Pfeffer, 1877). In the 1960s, Paul Green developed the first direct method to measure turgor pressure in cells, which would become the precursor of modern pressure probes (Green & Stanton, 1967; Green, 1968): an oil-filled microcapillary, which is connected to a pressure sensor and a moveable plunger, is introduced into a cell and the pressure necessary to maintain the liquid in the cell is measured as a quantification of internal turgor pressure (Husken et al., 1978; Cosgrove & Durachko, 1986). Based on these measurements, it has been estimated that turgor pressure in plant cells can build up to 10 bar. This value is extremely high, considering for example that 0.03 bar corresponds to a high blood pressure, and explains why plants are able to grow and even push their way through rocks or concrete.

Mathematically, the concept of growth driven by turgor was qualitatively formulated in 1877, and quantitatively summarized in 1965 by the Lockhart model (Pfeffer, 1877; Lockhart, 1965; Ortega, 1989; Tomos et al., 1989; Moulia & Fournier, 2009), which equates the expansion rate of a cell (dV/Vdt) to m(P − Y), where V is the volume of the cell, t is time, m is the extensibility of the cell wall, P is the turgor pressure, and Y is the minimal threshold of P below which the cell will not grow. This equation stresses the importance of the rheological properties of the cell wall in growth control. In the following we will, therefore, review its main features.

2. Defining the fundamental growth parameters at the cellular level: the role of the cell wall

Figure 1.

Anisotropy is controlled by the cortical microtubules. (a) Three-dimensional reconstructions of GFP-LTI6b meristems 72 h after a treatment with oryzalin: GFP-LTI6b meristems are imaged using confocal microscopy, and a projection of the cell surface is generated using the merryproj software (INRA-INRIA, France) (de Reuille et al., 2005). Note the formation of spherical primordia (i.e. a loss of anisotropy) and the presence of differential growth rates between the center of the meristem (lower growth) and the primordia (faster growth). Bar, 20 μm. (b) Close-up from (a). The shape of individual cell can be tracked and analysed. In the presence of oryzalin, walls tend to meet at 120° and cells tend to become hexagonal. Owing to the differential growth rates between the different domains of the shoot apex, each cell, being glued to its neighbors, displays some degree of anisotropic growth, even in the absence of microtubules. (c) Turgor pressure is the motor of growth. If the microtubules are oriented in a transverse orientation, the deposition of cellulose microfibrils should follow the same pattern and prevent wall expansion in a direction parallel to the microfibrils (i.e. promote a direction of growth in the axis of the cylinder created by the spirals of cellulose). If the orientation of the microtubules is more random, growth becomes isotropic. (d) A model: the cortical microtubules (CMT) located on the cytoplasmic side of the plasma membrane (PM) guide the movement of the rosette (CSC, cellulose synthesis complex) which synthesizes the cellulose microfibrils (CMF) in the cell wall, in a direction parallel to that of the CMT. (Reprinted from Emons et al. (2007) with permission from Elsevier.)

During growth, cellulose is deposited outside the plasma membrane, thus preventing the wall from becoming thinner and weaker. Because of their high tensile strength, cellulose microfibrils are the main determinants of cell wall stiffness. In addition, being structurally filamentous and often aligned in parallel arrays, cellulose microfibrils are also the main determinants of cell wall anisotropy: cells hardly grow in the direction parallel to the spirals of cellulose (Lloyd & Chan, 2004; Baskin, 2005; Cosgrove, 2005; Marga et al., 2005; Kutschera & Niklas, 2007; see next sections).

Between the microfibrils, two classes of polysaccharides, pectins and hemicelluloses, largely constitute the so-called cell wall ‘matrix’ together with structural proteins. Contrary to cellulose microfibrils, hemicelluloses are branched and this structural feature, together with their ability to bind cellulose, is thought to link the cellulose microfibrils into a cellulose–hemicellulose network (Somerville et al., 2004; Cosgrove, 2005). Other models envision hemicelluloses as spacers that prevent microfibrils aggregation (Thompson, 2005). Pectins control the stiffness of the cell wall, allowing or preventing the sliding of microfibrils (Willats et al., 2001; Cosgrove, 2005). More generally, through its noncovalent interactions with cellulose microfibrils, the matrix contributes to the mechanical properties and the dynamics of the cell wall.

While matrix polysaccharides are synthesized in the Golgi apparatus before being integrated into the cell wall, cellulose microfibrils are synthesized by cellulose synthases (CESA) located at the plasma membrane. Contrary to matrix polysaccharides which can diffuse into the cell wall, the newly synthesized microfibrils remain at the inner wall (Ray, 1967). The CESA proteins assemble into hexameric complexes forming particle ‘rosettes’ at the plasma membrane (as observed in the electron microscope after cryofracture). In Arabidopsis, the synthesis of the primary cell wall requires CESA1, CESA3 and CESA6 (Arioli et al., 1998; Fagard et al., 2000). Each CESA produces (1,4)-β-d-glucan chains that spontaneously bundle into a 4 nm wide microfibril. Each microfibril is long enough to run along the cell circumference several times (Cosgrove, 2005).

The turgor driven cell wall expansion is irreversible, and involves a slow reorganization of the microfibrils and matrix (also called creep). Schematically, the matrix loosens allowing the microfibrils to slide more easily and causing the wall to yield to forces generated by the internal pressure. Turgor pressure thus provides the mechanical energy for this polymer motion, while the loosening of the linkages between microfibrils controls the direction and rate of deformation. Several classes of molecules, namely expansin, pectin methylesterase, xyloglucan endotransglycolase/hydrolase, endo-(1,4)-β-d-glucanase and reactive oxygen species have been implicated in wall loosening. In the following paragraphs, we will discuss this topic more in detail.

The role of pectin methyl esterase (PME) in cell growth has recently received increased attention. Pectin is thought to represent one-third of all primary wall macromolecules (Willats et al., 2001). A defining aspect of pectins is that they are rich in galacturonic acid. When pectins are deposited in the cell wall, most (70–80%) of it is methylesterified (Ridley et al., 2001). The PMEs remove the methyl-ester groups resulting in stretches of acidic residues that can bind calcium and cross-link with other pectic chains. In this scenario, consistent with data obtained in pollen tube and hypocotyls, PMEs render the cell wall stiffer, leading to reduced growth (Jiang et al., 2005; Derbyshire et al., 2007; Rockel et al., 2008). By contrast, in the absence of calcium, de-methyl-esterification by PMEs is supposed to render the cell wall more fluid. De-methyl-esterification is also a prerequisite for pectin degradation by pectate lyase, which could further loosen the cell wall by opening an access to expansins and other hydrolytic enzymes in the cell wall (Cosgrove, 1999). The PMEs thus act as enzymes that make the cell wall able to become more or less stiff, depending on the action and presence of other effectors. To add a layer of complexity, PMEs belong to a large multigenic family and each isoform seems to have differences in their activities (Willats et al., 2001). Recent data are consistent with a loosening role of PMEs during the process of organ initiation in the shoot apical meristem. Notably, primordia position at the periphery of the shoot meristem correlates with the presence of de-methyl-esterified pectins, and primordium formation is inhibited when pectin de-methyl-esterification is impaired (Peaucelle et al., 2008).

Xyloglucans are the major hemicelluloses of primary cell walls (10–20% of the cell wall dry weight; Cavalier et al., 2008). Xyloglucans are the target of hydrolytic enzymes, namely the xylogucan endotransglucosylase/hydrolase (XTH). These have been implicated in both cell wall loosening and cell wall strengthening (Antosiewicz et al., 1997; Takeda et al., 2002; Cosgrove, 2005). Recently, using a reverse genetics approach, Cavalier and coworkers (Cavalier et al., 2008) obtained Arabidopsis lines with no detectable xylogucans. Results from micromechanical stress tests on these lines revealed a significant decrease in stiffness consistent with the contribution of xyloglucans to the mechanical properties of the primary cell wall. Surprisingly, the absence of detectable xyloglucans did not significantly alter the plant phenotype, thus challenging our understanding of the role of xyloglucans in the cell wall and more generally in growth and morphogenesis (Cavalier et al., 2008).

Xyloglucans and cellulose are also putative targets of endo-(1,4)-β-d-glucanases, also called cellulases. Three of them, including KORRIGAN, are membrane-bound and have rather been involved in cellulose formation (Nicol et al., 1998). Overexpression of PopCel1, a secreted endo-(1,4)-β-d-glucanases from poplar, in Arabidopsis increased wall extensibility and growth. Conversely, the reduction of endoglucanase expression resulted in reduced growth (Ohmiya et al., 1995, 2000; Tsabary et al., 2003). Although these enzymes appear to be important for wall loosening, their role in development is still poorly documented (Cosgrove, 2005).

In the past 10 yr, reactive oxygen species (ROS) have emerged as major wall loosening agents (Foreman et al., 2003; Liszkay et al., 2003; Schopfer & Liszkay, 2006). Contrary to the proteins presented so far, ROS do not display substrate specificity and can cleave any wall polysaccharides. Their effect on wall extension, directly by polysaccharide cleavage or indirectly through activation of signaling pathways, has been demonstrated in vitro and observed in vivo in actively growing tissues (Schopfer & Liszkay, 2006). Reactive oxygen species are produced by the NAD(P)H oxidase and cell wall peroxidase (Foreman et al., 2003). Interestingly, production of ROS, and their role in cell elongation, is auxin dependent (Schopfer, 2001; Schopfer et al., 2002) and might be subject to spatial regulation.

4. Cell wall anisotropy and the microtubular cytoskeleton

Turgor pressure is nondirectional, but the presence of aligned cellulose microfibrils in the cell wall translates turgor into anisotropic growth. Anisotropy is defined as the ratio of the two principal rates of growth (strain rates). Unless otherwise stated in the text, anisotropy will refer to the mechanical properties of the cell wall. Although the many determinants of cell wall loosening are still far from being completely characterized, the control of cell wall anisotropy is relatively well established and, in the most widely accepted concept, relies mainly on a single factor: the cortical microtubular cytoskeleton (Fig. 1). The morphogenetic role of the other major cytoskeletal element, the actin network, seems to be rather unspecific and/or indirect in most cells, although actin is crucial for the morphogenesis and development of cells undergoing tip growth (i.e. pollen tubes and root hairs).

Using the green alga Nitella, Paul Green first showed that colchicine disrupts cellulose alignment in the cell wall leading to isotropic (spherical) growth (Green, 1962). The major target of this drug turned out to be the cortical microtubules, which are attached to the plasma membrane in highly ordered arrays seemingly parallel to the cellulose microfibrils. Almost since their discovery, plant microtubule orientation has therefore been associated with the orientation of the cellulose microfibrils (Ledbetter & Porter, 1963; Emons et al., 2007). After the complex of cellulose synthesis, the CESA complex (also called the rosette), was located at the plasma membrane, it was hypothesized that cortical microtubules guide the movement of the rosette, thus explaining the colinearity between microtubules and microfibrils in growing cells (Heath, 1974; Mueller & Brown, 1980; Kimura et al., 1999; Fig. 1c,d). In this scenario, the polymerization of the stiff cellulose microfibrils would provide the force necessary for the movement of the rosettes, driving them forward. Using fluorescently labeled microtubules and cellulose synthase, Paredez and coworkers observed this movement in vivo and demonstrated that cortical microtubules guide the deposition of cellulose microfibrils in the cell wall (Paredez et al., 2006). In accordance with this scenario is the observation that microtubules are highly aligned perpendicular to the axis of cell expansion in highly anisotropic root cells. By contrast, cortical microtubules display rotary movements in very young hypocotyl cells and in the meristematic dome, consistent with the observation that growth is rather isotropic in those tissues (Chan et al., 2007; Hamant et al., 2008). It must be noted, however, that in certain cases, this colinearity is not observed, suggesting that the deposition of oriented cellulose microfibrils involves other levels of control (Himmelspach et al., 2003). Recently, an additional function of the cortical microtubules in the delivery of CESA complexes to the plasma membrane has been identified, showing a more complex role of the cortical microtubules in organizing cellulose synthesis at the cell cortex (Crowell et al., 2009; Gutierrez et al., 2009)

From a morphogenetic point of view, this provides a mechanism linking microtubules to the control of anisotropy. Indeed, in the footsteps of Paul Green, we observed a shift to isotropic growth in meristems treated with the microtubule depolymerizing agent oryzalin (Grandjean et al., 2004; Hamant et al., 2008; Fig. 1a,b) Interestingly, the geometry of meristematic cells after oryzalin treatment resembled that of soap bubbles in a two-dimensional (2D) froth: angles between adjacent walls in vertices tend towards 120° as in foams. More strikingly, the curvature of the walls was concave in cells having more than six walls and convex in cells having less than six walls as observed in 2D froths (Corson et al., 2009). Beyond the attractive homology between these two systems, this further demonstrates the crucial role of microtubules in anisotropy, as 2D froths are isotropic by essence.

Consecutively, the importance of the microtubules in controlling morphogenesis raises the question of the identity of the factors that control the organization of the microtubules. The cell wall itself influences microtubule organization: in a screen for mutants that are hypersensitive to oryzalin, Paredez an coworkers (Paredez et al., 2008) identified PRC1 and KOR, two proteins involved in cellulose synthesis, as regulators of microtubule organization. In a more extreme situation, when the cell wall is entirely removed during protoplasting, microtubules are unable to remain aligned, again confirming that the cell wall is directly or indirectly required for microtubule organization. The exact nature of the cell wall–microtubule dialog is still far from being completely elucidated.

A major aspect of cortical microtubules functions relates to their dynamics. Changes in the main microtubule orientation within a cell, seem to occur through the synthesis of microtubules in a new direction, which will then affect the orientation of the whole array by depolymerization and/or by reorientation of pre-existing microtubules (Lloyd & Chan, 2004). Katanin, a protein that cleaves the microtubules, appears to play an important role in this reorganization process. Mutations in the Arabidopsis katanin strongly affects anisotropy and result in compact, dwarf phenotypes, associated with reduction of cell length and increased cell width. This defect in the cytoskeleton also leads to problems in the cell wall which is weakened, owing to a decrease in the production of cellulose and hemicelluloses (Bichet et al., 2001; Burk et al., 2001). In katanin mutants, microtubules do not manage to organize in parallel arrays, and seem to remain connected to their nucleating site longer than in the wild-type (Burk et al., 2001). Using field emission scanning electron microscopy, a slight but significant alteration of microfibrils alignment could be detected in the innermost layer of the cell wall in the katanin mutant allele fra2, consistent with the role of microtubule organization in controlling cellulose deposition (Burk & Ye, 2002). Conversely, inducible overexpression of katanin in Arabidopsis favored the organization of microtubules into highly organized arrays. Longer induction led to complete microtubule depolymerization (Stoppin-Mellet et al., 2006). Based on these data, it was proposed that katanin activity helps to free the microtubules from their nucleating site by severing, thus affecting the frequency of their encounters at the cortex and allowing them to organize into arrays. Several other microtubule regulators, such as TONNEAU, CLASP1, MAP65 and MOR1, have been shown to control the organization of cortical microtubules by directly or indirectly modulating their dynamics (for a review, see Ehrhardt, 2008; Wasteneys & Ambrose, 2009).

III. From cells to organisms: a dialog between levels of organization

In the previous section, we described the basic components that determine growth rate and direction at a cellular scale. In this section, we will discuss the way the cells assemble and organize into tissues. The precise link between these two levels of organization has been a matter of debate in both the animal and plant fields. A range of observations seem to indicate that the cellular level is not an essential level of organization. For example, affecting cell division rate through transgenic approaches often does not dramatically affect shape, organogenesis and patterning, as a lack or excess of cell numbers is compensated by increased or reduced cell size (Vernoux et al., 2000). The tonneau and tangled mutants exhibit random cell division planes, while generating organs with correct identity, leaving patterning apparently unperturbed (Smith et al., 1992; Traas et al., 1995; Cleary & Smith, 1998). These results support the so-called organismal theory of development (Kaplan & Hagemann, 1991; Kaplan, 1992) which revives an older aphorism stating that ‘it is the plant that forms the cells, not the cells that form the plant’ (de Bary, 1879). This statement may be opposed to the cell theory, which was first formulated by Dutrochet (Dutrochet, 1824) and which gives a central role of the cell in morphogenesis. While these different views on the mechanism of morphogenesis (one top-down, the other bottom-up) have been actively debated (Meyerowitz, 1996; Tsukaya, 2003), they can be now reconciled by the concept of complex systems and systems biology.

A complex system is a set (or a whole) of entities that interact according to simple local rules. These interactions lead to the emergence of properties at a higher level of organization that cannot be deduced from the local properties. As an example, we discussed earlier how the instability of single microtubules can explain the formation of microtubule arrays. Biological organisms are typical examples of complex systems where the individual cells locally interact (e.g. via the exchange of simple molecules) to form organs and individuals with specific shapes and global properties. Viewing a plant as a complex system, morphogenesis depends on both genetic information (bottom-up) and a supracellular feedback exerted on individual cells (top-down). Systems biology offers a framework to analyse living complex systems. It analyses processes such as plant development at different scales, describing not only the properties of individual cells but also their interactions (for a recent review see Grieneisen & Scheres, 2009). After having described the basic cellular properties, we will now describe the interactions that lead to the formation of organized structures.

1. Integration of growth by hormone signaling

Several recent articles have reviewed how a range of molecular signals can generate patterns of growth (Bowman & Floyd, 2008; Lewis, 2008; Wang & Li, 2008). As this does not fall into the main scope of this review, not all of them will be discussed in detail here. We will simply provide focus on auxin and gibberellins because of their documented implication in cell wall dynamics and mechanical properties.

Auxin was first identified as a growth factor that enables cell expansion in coleoptiles during phototropism, and later gained the status of a universal inducer of plant cell elongation (Darwin, 1880; Boysen-Jensen, 1913; Went, 1928; Dharmasiri et al., 2005). How auxin induces cell elongation is still largely under debate but it seems that the hormone can influence wall properties in several ways. There is accumulating evidence in support of the acid growth theory, which proposes that auxin stimulates proton pumps, leading to the secretion of protons into the cell wall and a decrease in apoplastic pH, which in turn activates wall-loosening agents (Hager et al., 1971; Rayle & Cleland, 1992; Kutschera & Niklas, 2007). Among many supporting observations of this model, it is notable that the pH of the cell wall of growing cells is typically between 4.5 and 6 (for a review see Hager, 2003). Auxin also indirectly affects cell wall dynamics by inducing the expression of several wall loosening proteins. For example, the expression of several XTHs, expansins, endo B1,4 glucanase, extensins and pectinesterases, is repressed in the axr3-1/iaa17-1 mutant (Overvoorde et al., 2005).

Growth factors not only affect the rheological properties of the cell wall, but also control its anisotropy. Cortical microtubule orientation switches to transverse in the presence of auxin, gibberellins (GAs) and brassinosteroid consistent with the role of these hormones in promoting cell elongation (Ishida & Katsumi, 1992; Baluska & Barlow, 1993; Zandomeni & Schopfer, 1993). Conversely, abscissic acid and ethylene induce an oblique orientation of the microtubules. The cascade of events leading these hormones to affect the orientation of the microtubules is unknown, although we have more clues in the case of GAs.

Mutants in rice or Arabidopsis in GA synthesis or signaling usually are dwarf because of internode elongation defects. For example, the lue1 dwarf mutant has been identified in a screen for misexpression of GA20 oxidase, a key protein of GA biosynthesis. Interestingly, the lue1 mutant is allelic to katanin mutants. Complementation of lue1 with the katanin WT gene restored a normal GA20 oxidase expression, but GA applications on the mutant did not rescue the phenotype (Bouquin et al., 2003). These results identify the presence of katanin dependent crosstalks between GA biosynthesis pathways and cortical microtubules organization. Interestingly they suggest a feedback between signals operating at tissue level and local cellular components.

2. Integration of growth by mechanical forces

A major aspect of cell wall expansion is that each cell wall is glued to its neighbor by a pectic middle lamella. This affects morphogenesis on two levels. First, the presence of the middle lamella prevents cell migration, and morphogenesis at the tissue scale is thus essentially the result of cell division and cell elongation. Second, at the cell level, the growth of a given cell depends on the rate and direction of cell wall expansion of its neighbors From a morphogenetic perspective, this ‘glue’ is the source of a mechanical dialog between the cells (and their gene network) and the overall shape of organs and tissues (a multicellular object under mechanical constraints). For example, the increased extensibility of the cell wall in a region (typically a primordium in a meristem) will increase the growth rate locally and, because the cells are glued to each other, this will generate mechanical constraints in the surrounding cells (e.g. in the boundary domain of the meristem). A major aspect of these mechanical signals is that they can be transmitted instantaneously at relatively long distances, and thus contribute to the coordination of growth (Lintilhac, 1984). In addition, like chemical gradients, they can provide directional and positional information.

IV. Biomechanics: some concepts and approaches

The presence of internal forces is a universal feature of all growing systems in all kingdoms (Hejnowicz, 1997; Sharon et al., 2002; Moulia & Fournier, 2009). A classical and basic experiment demonstrating the presence of patterns of forces in a tissue is to cut lengthwise the bottom of a dandelion stem. The stem will immediately curl upwards because of the relaxation of longitudinal tension between the outer and the inner tissues, and fitting the pieces back together will require work, thus energy. Conversely, this also shows that before splitting the stem, the internal forces balanced themselves out.

1. Stress and strain

When considering a force acting on a given material, one immediately realizes that the amount of force necessary to produce a deformation will depend on the nature of the material (its stiffness) and its geometry (e.g. its thickness). To analyse the internal forces within a tissue, the concept of stress has been formulated. Mathematically and physically, stress (often noted σ) can be defined as a force per unit area upon which the force is acting. As it is defined per unit area, stress does not depend on the geometry of the material, and really accounts for the mechanical properties of the material (Fig. 2a).

Figure 2.

Investigating the role of physical forces during development. (a) Defining stress and strain: Stress (σ) is defined as a force (F) divided by the surface (A0) on which the force acts. In the example shown, the (isotropic) material is under unidirectional tension, and the length increases from L0 to L. This deformation is measured by the strain (ε) which is defined as the ratio between L − L0 and L0. Hooke’s law for a spring (F = −k(L − L0)) relates the force F to the deformation via k, the stiffness of the spring. Similarly, Hooke’s law for materials (σ = Eε) relates the stress σ to the strain ε via E the Young modulus of the material. (b) Stress is measured by the strain it produces. The replica technique is used to track cell shapes at the meristem surface: at successive time points, a scanning electron microscopy (SEM) image of the meristem surface is taken at two different angles and the strain rate of each cell is quantified: low (blue) to high (red). The main direction of strain is also represented by a cross arm, the length of which is proportional to the corresponding principal strain rates. Note the high strain rate in the emerging primordium P2 and the tangential strain rate in the boundary domain next to primordia P3 and P4 (from Kwiatkowska & Dumais (2003) by permission of Oxford University Press). (c) Analogies are used to investigate how physical forces could generate patterns in biological systems. In this example, the venation pattern in leaves (left) resembles cracks in drying gel (right). In a stretched medium, the presence of individual cracks is not stable thermodynamically and cracks tend to intersect. The cracks in drying gel thus exhibit loops, following a pattern close to that of leaf vasculature (from Couder et al. (2002), with permission of The European Physical Journal (EPJ)). (d,e) Modeling is used to test the plausibility of supracellular morphogenetic processes based on the mechanics of individual cells. (d) Walls are represented as springs to model growth in two dimensions: as growth occurs, the resting length (L0) of the spring increases (red). The orientation of the microtubules can then be introduced in the model to monitor cell-based anisotropic growth, the stiffness of the spring perpendicular to the microtubules being lower than that of the spring parallel to the microtubules. (e) Assuming microtubules (red lines) align along main stress directions, a circumferential microtubule pattern is obtained at the periphery of the meristem. Decreasing the stiffness of the walls in a domain of the meristem (light green) leads to the emergence of a bump (a virtual primordium) and induces a stable and tangential microtubule alignment at the boundary domain, as the primordium emerges (from Hamant et al. (2008) reprinted with permission from AAAS). (f,g) Stress patterns are modified in order to establish a causality link between stress and cell behavior. To investigate more directly the role of physical forces in the meristem, a shoot apex (e, sunflower; f, Arabidopsis) is compressed in a microvice and the pattern of stress is modified (from Hernandez & Green (1993); reprinted with permission of the American Society of Plant Biologists) the main direction of stress becoming parallel to the main strain. Using a transgenic line expressing the GFP-MBD construct (marking the microtubules), we can analyse the effect of these compressions on the dynamics of the microtubules in the meristem. Bar, 20 μm.

Experimentally, internal forces in a tissue can only be measured indirectly, and the effect of stress on a body such as a tissue or an organ can be quantified as a deformation called strain (e.g. the curling response in the dandelion experiment). Stress in cells is therefore measured by the strain it produces. As described by Hooke’s law (see Fig. 2a), the relationship between stress and strain relies on the stiffness: when a stress is applied to an object and the resulting strain is measured, the slope of stress–strain plot is the stiffness (Wozniak & Chen, 2009). The mean elastic stiffness of the cell wall is called its Young’s modulus. Atomic force microscopy, which allows the acquisition of such stress–strain plots, is a method of choice to measure the stiffness (and Young’s modulus) with a subcellular resolution (Bao & Suresh, 2003). This tool, combined with live imaging, currently emerges as a way to quantify the mechanical properties of individual cells and correlate the measurements with changes in cell shape and with the dynamics of molecular effectors, such as the cytoskeleton (Chaudhuri et al., 2009).

To measure strain in a tissue, one has to track the position of defined landmarks (typically cell vertices in plants) over time. For example, in the simplest form of growth (2D planar growth), the magnitudes of the two principal rates of growth and their direction is sufficient to fully describe growth. This approach is included in the more general branch of classical mechanics called kinematics (Fiorani & Beemster, 2006; Kwiatkowska, 2008). In a coordinate system, displacement vectors can be drawn. The rate of displacement can also be drawn as velocity vectors, and the rate of change in velocity as acceleration vectors. In plants this approach has been successfully used to describe the growth of individual leaves and other organs (Erickson, 1976; Fiorani & Beemster, 2006; Chavarria-Krauser et al., 2008). In the shoot apical meristem, epoxy resin casts have been made at different time-points to track the very early stages of organ initiation with high spatial resolution (Williams & Green, 1988; Kwiatkowska & Dumais, 2003; Fig. 2b). This approach notably identified a shift in the strain rate of the boundary domain that separates the young organ from the meristem (Kwiatkowska & Dumais, 2003; Kwiatkowska, 2008). Live imaging techniques have also been developed to analyse growth. In particular, using fluorescent proteins or dyes and water immersion objectives, optical sections through a growing meristem can be obtained by confocal microscopy at different time-points (Grandjean et al., 2004; Reddy et al., 2004). The development of three-dimensional (3D) reconstruction software remains an essential step to successfully quantify and analyse the growth patterns obtained via these techniques (de Reuille et al., 2005; Routier-Kierzkowska & Kwiatkowska, 2008; Oates et al., 2009; Fig. 1a,b, 2b).

2. Calculating stress patterns

If it is obvious that forces operate in the growing tissues, what is their intensity and direction? Continuum mechanics, which assumes a homogeneous distribution of the material properties, has been widely used to calculate stress patterns in tissues (Williamson, 1990; Dumais, 2007; Kutschera & Niklas, 2007; Hamant et al., 2008). In plant cells, as already discussed, stress is caused by turgor pressure and mainly depends on the rheological properties of the cell wall. In addition, knowing that cell migration does not occur in growing plant tissues, the approximations behind continuum mechanics can be applied to most plant tissues*.

In this framework, a general model of growth, based on an analogy with pressure vessels, has been proposed to explain shapes in plants. Knowing that the external epidermal wall can be 5–10 times thicker than the walls of inner cells, one can assume that this thicker wall is limiting for growth. From a biomechanical point of view, the presence of a thicker outer epidermal wall implies that internal tensile forces build up and increase the tension in the outer epidermal cell wall (Williamson, 1990; Kutschera & Niklas, 2007; Hamant et al., 2008). Consistent with this, while in water, the peeled epidermis of a coleoptile segment contracts while the rest of the segment expands, indicating (as in the dandelion experiment) that the inner tissues are in a state of compression while the epidermis is in a state of tension (Kutschera et al., 1987; Schopfer, 2006). By analogy, the distribution of stresses in a plant organ can therefore be calculated from a continuum mechanics perspective based on the physics of pressure vessels (Flugge, 1973). Typically, in a cylindrical stem, the direction of principal stress will be circumferential. Normally this would cause the tissue to swell and lead to balloon-like organs. The main strain, however, is perpendicular to the main stress. This can be attributed to the fact that the cells themselves, and particularly the cell walls, have anisotropic properties, as the cellulose microfibrils are perpendicular to the main stress orientation. Thanks to the cellulose microfibrils, cells resist to the main circumferential stress, allowing the residual axial stress to drive growth along the axis of the cylinder (Green, 1980; Hejnowicz, 2000).

Continuum mechanics can even become a source of inspiration to design experiments in which geometrical patterns are generated on physical objects and then compared with the natural patterns observed in plants. The analysis of venation in leaves offers a striking example: taking advantage of the analogy between cracks in a drying gel and venation patterns, Couder and coworkers (Couder et al., 2002) suggested that supracellular constraints can determine the geometry of the vasculature (Fig. 2c).

The morphogenetic output as a result of physical interactions between cells is difficult to predict on a purely intuitive basis; therefore, modeling approaches can be extremely helpful in analysing how the mechanical properties of individual cells impact morphogenesis at a global scale (for recent reviews, see Geitmann & Ortega, 2009; Oates et al., 2009).

Two types of modeling approaches are currently being developed. First, growing cells in a 2D tissue are represented as vertices connected by springs. Growth is implemented by increasing the resting length of the spring. Anisotropy can also be implemented by modulating the stiffness of the spring in certain walls (Fig. 2d,e). An extension of this approach is the use of finite elements. Instead of using springs between vertices, the finite element modeling generates several small elastic domains on each cell surface. This method is particularly adapted to model growth in three dimensions (Coen et al., 2004).

One great advantage of these modeling approaches is that they can reproduce morphogenetic events based on individual cell response, integrate biochemical or genetic factors and predict cell responses to stress. For example, in a disk-shaped tissue where regions at the edges grow faster than the inner regions, stress is accumulated and is released by ruffling the edges. The presence of such structures in nature can thus be potentially explained by this peripheral growth hypothesis (Sharon et al., 2002). Conversely, the absence of ruffles in other systems can also imply that the cells are able to sense the stresses and carefully control growth to maintain a flat surface. This has been shown in Antirrhinum leaves which are planar in wild-type and wavy in the cincinnata mutant because of a delay in growth arrest at the margins of the leaf (Nath et al., 2003). Here we must underline that the fact that a mechanical simulation of growth matches the observed growth demonstrates that the hypotheses formulated in the model are plausible; however, it does not prove that these hypotheses are the only valid ones.

In plants, the putative role of stresses in morphogenesis has been addressed in several systems. In this context the positioning of organs on a stem (phyllotaxis) has received much attention. Phyllotaxis is largely defined at the shoot apical meristem where all organs are initiated. It involves a set of complex changes in shape in highly dynamic and stereotypic patterns, making it a highly suitable system to analyse the biophysical basis of pattern. Geometrical, mechanical and biochemical models have been proposed to explain phyllotaxis. The most documented and supported biochemical model to date states that a young organ emerges at sites where auxin maxima arise, generated by the active cell to cell transport of auxin (Reinhardt et al., 2003b; Heisler et al., 2005; Jönsson et al., 2006; de Reuille et al., 2006; Smith et al., 2006; Stoma et al., 2008; Bayer et al., 2009). As the young growing organ draws away auxin from its vicinity, no new maximum can form near to it, thus leading to a regular spacing of the primordia. An alternative, not necessarily mutually exclusive hypothesis has been proposed by Green and colleagues. In this so-called buckling theory, the generation of a local outgrowth will affect the position of a neighboring one via mechanical constraints. This idea is based on the observation that presence of a bump in a flat sheet prevents bending in its immediate proximity. Such a biophysical version of the inhibition field could, in principle, also produce a regular spacing of outgrowths (Green, 1980; Dumais & Steele, 2000; Shipman & Newell, 2005; Dumais, 2007).

This mechanical theory of phyllotaxis has been widely discussed theoretically, but only partially addressed experimentally. To test whether organ position could depend on a mechanical signal, the presence of compressive forces within the meristem has been investigated through microsurgical manipulations. Radial cuts were performed into the surface of a sunflower capitulum and these induced a wide gap in the center of the meristem. By contrast, the gaps remained closed in the organ production domain (Dumais & Steele, 2000). From the gaping pattern, the presence of compressive forces in the peripheral zone of the meristem was deduced, a necessary condition for the buckling theory.

To test the buckling theory more directly, several tools have been created, aiming at compressing shoot apices for prolonged amounts of time (Fig. 2f,g). In these experiments, the main direction of stress is altered and an ectopic folding (or buckling) of the tissue is induced. By constraining a Graptopetalum meristem between two glass chips, Paul Green managed to induce the production of an ectopic leaf (Green, 1999). A similar experiment performed on sunflower capitulum induced patterning defects. Notably, the shape and identity of organs were altered (Hernandez & Green, 1993). Formation of ectopic primordia was also induced by the local increase in cell-wall extensibility through local application or expression of expansin (Fleming et al., 1997; Pien et al., 2001). More recently, the use of transgenic lines with altered PME activity also led to altered phyllotactic patterns (Peaucelle et al., 2008). From a biomechanical perspective, it would be extremely interesting to be able to uncouple the mechanical role of PMEs from their biochemical effect to evaluate the contribution of mechanical forces in phyllotaxis. Together, these experiments support the view that phyllotaxis largely depends on physics and that physical forces directly provide information for patterning (Green, 1980; Fleming et al., 1997; Pien et al., 2001; Peaucelle et al., 2008).

If we accept that the observations mentioned above strongly argue in favor of a physics-based control of patterning at the shoot apex, does this exclude a mechanism based on biochemistry? Several arguments seem to show that both actually operate in parallel. First, several computational models show that it is possible to generate a phyllotactic pattern that is only based on gradients of auxin and auxin transport (Reinhardt et al., 2003b; Heisler et al., 2005; Jönsson et al., 2006; de Reuille et al., 2006; Smith et al., 2006; Stoma et al., 2008; Bayer et al., 2009). Second, the depolymerization of microtubules in the meristem completely abolishes the anisotropy of the cell wall but does not affect phyllotaxis at least for 2–3 d (Grandjean et al., 2004; Hamant et al., 2008), and patterning continues despite the fact that the system is physically perturbed. Finally, local ablation of the epidermis only has local effect on growth and does not alter phyllotaxis (Reinhardt et al., 2003a). These results show that phyllotactic patterns are very robust and suggest that auxin and mechanical signaling are highly coordinated (Newell et al., 2008). The characterization of the distribution of auxin following short-term physical perturbations would certainly help in understanding the link between mechanics, auxin-based patterning and morphogenesis. More generally, the analysis of the molecular actors of cell morphogenesis and their response to mechanical constraints should unravel the cross-talks between biochemistry and mechanics in controlling plant development.

4. A microtubules-based feedback loop

Given the strong implications in terms of morphogenesis, the idea that stress can orient microtubule orientation has been thoroughly theorized in the past (Green & King, 1966; Williamson, 1990). Previous studies on protoplasts showed that centrifugation induces a reorientation of the microtubules parallel to the direction of centrifugal force (Wymer et al., 1996). Using live imaging and modeling approaches, this issue was recently re-examined in the shoot meristem (Hamant et al., 2008).

First, cortical microtubule orientations were analysed in living meristems: microtubules display chaotic orientations at the tip of the meristematic dome, circumferential orientations at the base of the meristem and on the stem, and a stable and tangential stress orientation in the boundary domain that separates the young organ from the meristem. This stereotypical pattern is consistent with previous data obtained on fixed meristems (Sakaguchi et al., 1988; Marc & Hackett, 1989).

When microtubules are depolymerized with oryzalin, the sharp crease at the boundary domain – the domain where microtubules display a stable, supracellular and tangential alignment – is smoothed or even disappears, supporting the view that microtubule orientation at the boundary controls cell wall anisotropy, hence crease formation (Green, 1980; Jesuthasan & Green, 1989).

Assuming that the meristem surface is under tension (and following a pressure vessel analogy, see previous section), the patterns of stress orientation can be predicted based on meristem shape. In particular, such a model predicts random and/or unstable stress direction at the meristematic dome, circumferential stress direction on the stem and at the periphery of the meristem, and stable and tangential stress direction in the boundary domain. The strong correlation between stress and microtubule orientation supports a model in which microtubules orient along main stress directions. To test this concept, we generated a mechanical model in which this rule was implemented. The model not only reproduced the observed microtubule pattern in a real meristem, but also reproduced morphogenesis in wild type, pin-formed and oryzalin-treated meristems. Experimentally, the compression of meristems (and the predicted shift in stress direction parallel to the stretch) led to a noisy but significant reorientation of the microtubules parallel to the main direction of stress (Fig. 2g). Last, cell ablations on the meristem surface induced circumferential microtubule orientations, matching the predicted induced stress patterns. To test whether a biochemical signal could induce such a response, cell ablations were performed in two cells separated by one cell. In a biochemical scenario, the diffusion of a molecule from each ablated cell should not produce any directional information in the cell that neighbors both ablated cells; by contrast, in a mechanical scenario, circumferential stress direction in both ablated cells should strengthen the stress direction in the cells neighboring both ablated cells. The mechanical scenario was validated experimentally, thus supporting a model in which microtubules control anisotropy and shape, and shape in turn controls microtubule orientation via the pattern of stresses (Fig. 3). Interestingly, the presence of such a feedback loop also implies that stress controls cell division by orienting the pre-prophase band and thus the plane of division. By controlling the orientation of crosswalls this feedback mechanism also influences the physical properties of the tissue. To some extent, these findings are consistent with the tensegrity (tensional integrity) concept, in which tension controls cytoskeleton behavior and cell shape (Ingber, 1993; Pickett-Heaps et al., 1999).

Figure 3.

A working hypothesis for the control of morphogenesis at the shoot apex. On one side, cortical microtubules control cell wall anisotropy and on the other, auxin and other effectors control cell wall expansion rate. These two variables, namely anisotropic growth and growth rate, are sufficient to explain all morphogenesis. They also are the source of a pattern of mechanical forces, that could be perceived by several proteins or molecules involved in mechanotransduction. While the presence of a mechanical feedback loop on the microtubules has been supported by several studies, it is still unclear what impact mechanical forces could have on auxin synthesis, transport or signaling.

V. Sensing forces: some clues on the molecular basis

If mechanics can serve as a signal, in addition to its role in driving growth, how is it sensed in plants?

The binding of a specific ligand to a specific receptor is the basis of current molecular models for hormone or metabolite signaling. At first sight, sensing stress seems very different but, as for chemical signaling, it must somehow rely on protein conformation change. The activity of proteins depends on their 3D conformation. It is well known that temperature, pH or ligand binding can change protein conformations. A series of experiments performed in the late 1990s demonstrated that mechanical forces can also deform or unfold certain protein domains (Kellermayer et al., 1997; Rief et al., 1997; Tskhovrebova et al., 1997; Oberhauser et al., 1998; Vogel & Sheetz, 2006; Oates et al., 2009). This has been demonstrated for extracellular matrix proteins in animals, as well as cytoskeletal proteins and G-protein-coupled receptors (Bao & Suresh, 2003; Kumamoto, 2008). Thus, in the presence of mechanical forces, the conformation of a binding site can change, and switch on or off a signaling cascade (for a detailed review, see Vogel & Sheetz, 2006).

The wall, being at the forefront of the mechanical properties of plant cells, represents a very good candidate to mediate the mechanotransduction. Several potential wall-linked sensors, such as the wall-associated kinases (WAKs) or the lectin receptor kinases have been identified and could be involved in mechanoperception (for a detailed review, see Humphrey et al., 2007).

Several lines of evidence suggest that the cell wall might not be the only point of entry for a mechanical signal, but that the plasma membrane might also play a role. The experiments by Wymer et al., showing microtubule rearrangements upon centrifugation (Wymer et al., 1996), were performed on protoplasts in which the cell wall was dramatically altered and largely removed. Second, plasma membrane linked, stretch-activated channels have been identified in every kingdom and strongly suggest a major role of ionic transport in the mechanical response (Hamill & Martinac, 2001; Kung, 2005; Zonia & Munnik, 2007). In plants, stretch-activated channel activities have been recorded in various conditions but the genes coding these channels remain to be identified in most cases (Nakagawa et al., 2007; Zonia & Munnik, 2007; Haswell et al., 2008).

The cytoskeleton is also likely to play a key role in mechanotransduction. Data from Drosophila show that tension at the epithelium, locally induced by myosin activity, increases proliferation and regulates growth, providing a mechanical feedback for morphogenesis (Nelson et al., 2005; Shraiman, 2005; Hufnagel et al., 2007; Lecuit & Lenne, 2007; Wang & Riechmann, 2007; Oates et al., 2009). The molecular cascade involving myosin II activation is well described (for a detailed review see Lecuit & Lenne, 2007; Wozniak & Chen, 2009). Although it seems to react to mechanical signals, there is no evidence that stress orientations are directly sensed by the cytoskeleton during growth in plants. In animals, integrins serve as mechanical links between the cytoskeleton and the extracellular matrix and they have been shown to act as mechanoreceptors (Kumamoto, 2008). Although plant genomes do not have obvious integrin homologues, the RGD motif found in certain plant proteins could give them an integrin-like function (Jaffe et al., 2002). In particular, RGD peptides are able to disrupt the Hechtian strands that link the cell wall to the plasma membrane and which contain actin and microtubules (Canut et al., 1998; Mellersh & Heath, 2001).

Several target genes are regulated downstream of the mechanical signal. Transcriptomic analysis have been undertaken to identify such genes (Moseyko et al., 2002; Leblanc-Fournier et al., 2008). Although it is difficult to establish how mechanical stress-specific they are, it is believed that 2.5% of Arabidopsis genes are upregulated in response to touch (Braam, 2005).

VI. Conclusion

The integration of molecular and cell biology with physics is crucial to elucidate the dialog between the genes and the organism, and to understanding development. Modeling is becoming a widely used tool to investigate this dialog, notably because it can help investigate the robustness of developmental hypothesis and test the plausibility of feedback loops – a defining aspect of post-genomic biology (Jaeger et al., 2008). In other words, systems biology helps us to understand how the patterns of morphogenesis are highly reproducible despite the noise of individual developmental responses. Analysing the interplay between this flexibility (the noise) and this robustness (the patterns) is also crucial to understanding development during evolution and explaining why certain patterns have been selected to the detriment of others.

Footnotes

*

Note that this is often not true for animal tissues as cell migration can occur and the amount of stress experienced by animal cells not only depends on the stiffness of their direct environment (the extracellular matrix or neighboring cells) but also on the contractile activity of each individual cell (Wozniak & Chen, 2009).

Acknowledgements

We thank Françoise Monéger and Teva Vernoux for their comments. We also acknowledge stimulating discussions before this publication with participants to the ‘Biomechanics in Growth Event’ held in Bristol in March 2009 and to Jacques Dumais’‘Lectures on biomechanics’ held in Bern in June 2009. We acknowledge research support from a bilateral grant from the Ministry of Science and Higher Education, Poland and the Institut National de la Recherche Agronomique, France.

Baluska F, Barlow PW. 1993. The role of the microtubular cytoskeleton in determining nuclear chromatin structure and passage of maize root cells through the cell cycle. European Journal of Cell Biology61: 160–167.

Green PB, King A. 1966. A mechanism for the origin of specifically oriented textures in development with special reference to nitella wall structure. Australian Journal of Biological Sciences19: 421–437.